How beautiful is mathematics?

From James Gallagher of the BBC:

Mathematicians were shown “ugly” and “beautiful” equations while in a brain scanner at University College London.

The same emotional brain centres used to appreciate art were being activated by “beautiful” maths.

The researchers suggest there may be a neurobiological basis to beauty.

The likes of Euler’s identity or the Pythagorean identity are rarely mentioned in the same breath as the best of Mozart, Shakespeare and Van Gogh.

The study in the journal Frontiers in Human Neuroscience gave 15 mathematicians 60 formula to rate.

Euler’s Identity is a particular favorite of mine, and indeed:

The more beautiful they rated the formula, the greater the surge in activity detected during the fMRI (functional magnetic resonance imaging) scans.

…To the untrained eye there may not be much beauty in Euler’s identity, but in the study it was the formula of choice for mathematicians.

Oh, and this:

In the study, mathematicians rated Srinivasa Ramanujan’s infinite series and Riemann’s functional equation as the ugliest of the formulae.

For the pointer I thank Joanna Syrda.


I'm not a mathematician, but I've always found Euler's formula to be beautiful. Glad that I'm not as much of a weirdo as I thought!

I am a mathematician, and yes, Euler's formula is beautiful...

A third vote for Euler - the most beautiful thing I saw in school.

Sad, sad man, you need to get out more.

Personally, I am awe-struck by "Euler's identity" (or would be if I understood it as well without as with lots of caffeine and concentration), but it may be far from Euler's best work, and I think it is a little too proto-modernist for many. When pictured as a set of logical steps, it can appear semi-fraudulent, like it was put together by a very clever, and possibly not vey honest, person with a very small mathematical vocabulary. (Euler himself was extremely honest, I am sure). In art terms, it can be viewed more as a Russian formalist fashionable painting (i.e., one small moral step above a forgery) than as a Renaissance image, divinely colored and shaped by genius. Put otherwise, more variations on "twinkle twinkle little star" than "Ave Verum Corpus" (good Mozart versus very very good Mozart).

It is, however, the only piece of Euler's work that made it into the Simpsons twice. And isn't that the true enduring measure of success?

No doubt Euler's Identity ( a special case of Euler's function, is the sexiest formulae.

Want more of that? Why Beauty Is Truth: The History of Symmetry by Ian Stewart

Pleasing might be a better word than beauty here. Any good musician has the same feeling or sense of appreciation when he/she hears a particularly well played, composed, created piece of music, even very short. I venture that a boxer would have a similar sense in seeing a fellow boxer land a perfect punch or when they do it themselves (even on focus mitts, because landing a perfect punch in a match is a rare occurrence). I know little about any other subject but might not plumbers, diamond cutters, coders, hookers, lawyers, economists, politicians, con-artists, almost anyone, experience the same sense of appreciation or whatever when seeing what they have learned to see as well-constructed, well-done, or whatever, and might not that have some reflection in brain functioning? Why does it matter? Well, for a musician or boxer, it tells you when you are doing something right. Do it again or try.

As you can read on Wikipedia, the Riemann functional equation becomes the nicely symmetric xi(s) = xi(1-s) when expressed in terms of the xi function. This form is so simple and the xi function so specific to one area of study that it probably wouldn't even be recognized as the Riemann functional equation when displayed without context, so this probably wasn't the form they used. I certainly hope that the equality of the Euler product and infinite series forms of the Riemann zeta function was tested and found to be very beautiful.

Th problem is the Euler formula doesn't do anything. It's more of an identity than a formula

Bach's Cello Suite No. 1 doesn't do anything either.

I once had a nonlinear problem that I took to a mathematician in case I'd overlooked something. He said "That's not interesting, it's merely difficult".

Post-Tate, Riemann's functional equation is amazingly beautiful. That Gamma function isn't just there randomly. As always, Terry Tao is a good place to read about it:

Maybe, they just showed the wrong form of the equation :).

They did indeed show the wrong form -- well, they showed the form using zeta rather than xi(s) = xi(1-s) where xi has the gamma factors built in.

anything less

spacetime = matter&momentum

Spacetime = Wife and kids asleep at 10pm.

Euler's identity is one example of the beauty of doing mathematics with complex numbers. The world of the real numbers is so poor and, uh, one dimensional in comparison.

Euler's identity is a special case of the fact that multiplying any complex number z by e^(x*2pi*i), where x is in [0, 1) results in a rotation of z by x*360 degrees about the origin. (For Euler's identity, just rotate the number 1 + 0i by 180 degrees about the origin to get -1 + 0i.) So, using complex numbers, one can rotate any shape, or set of points, in the complex plane by a simple multiplication. I've always loved this aspect of complex numbers, and how they allow geometry to be expressed via algebra.

Made practical in application with the Sit and Spin.

Ah, complex numbers - quaternions for dummies.

How long have we known the Golden Ratio? I seem to recall it being a Greek concept, but I may be mistaken. Pleasing images and sounds would certainly fall under evolutionary pressure. Culture and biology are not that far apart.

The math is beautiful. You're just too much of a philistine to appreciate it.

If only the two cultures mixed. I'd love to throw that one out there.

So now we know why Middle East peace is so elusive.

My favorite is the Townshend Equations:


My marriage. Q.E.D.

I call that a bargain.

Best I ever had.

Darn, hit submit too soon.

Anyway, that expression evaluates to

0. 0001 0002 0004 0008 0016 0032 0064 0128 0256 0512 1024 . . .

nice one, although it gets more complicated after 4096

Yeah, the field after that (16384) overflows into the previous one. You can add more 9's to make the fields wider.

It works for fewer 9's as well. 1/98 has fields two digits wide.

1/8 technically works, too. It's just that all the powers of 2 add up to 0's past the third decimal place so it looks, depending on your viewpoint, either a) extremely uninteresting as it's all 0's, or b) extremely interesting as it's all 0's.

Well, I guess each decimal place past the third is 9, making 1/8 = 0.1249999999..., which is great fun to tell people at parties is the same as 0.125.

i am out of math shape. what is the series ?
1/(x-2) = sum 1/(2^nx^n+1)

I had no idea there is so much fondness for Euler's Identity. My past several years of work have mostly been working out generalizations of potential flow around circles to more general geometries, so there I am on a daily basis working through conformal mappings onto unit circles and the implications that follow. Cauchy residues, Blasius' Theorem, and such. The complex plane is a fun place to work, the way real and imaginary parts combine as entwined, non-independent aspects of analytical functions, but I never gave Euler's Identity much thought beyond "Gee, that's neat." Most beautiful formula? I don't see it.

What I have delighted in in the complex plane is analytic continuation. Conjugation is non-analytic, a show stopper that takes all your tools away. But on the unit circle, the conjugate is the inverse, so conjugation can be replaced with inversion if the contour of interest is mapped to the unit circle. I have found beauty in that.

If you don't see it, you haven't thought about it enough.

its a short equation using the five most fundamental numbers in a relation using the three most fundamental operators

Maybe I dont remember math well, but isnt it three most fundamental numbers (pi, e, i) and five operators (+, *, ^, sin, cos)?

dude. zero is a number. 1 is a number.
you could include sin and cos if you want
e^ipi = cos(pi) + i sin(pi)

Brian, as Indicated I spend a lot of time thinking about the complex plane, filling 11"x17" quad sheets with equations that often have combinations of e, i, and pi in them, formatting the good stuff with LaTex, testing concepts with MATLAB and programming production code in FORTRAN (still the lingua franca of computational fluid dynamics). Hundreds of hours a year over the last eight years. So more time alone won't help me. Give me a paragraph describing the beauty of Euler's Identity. What am I missing?

That's the same thing people tells me when I dismiss contemporary art or "performance".

> The researchers suggest there may be a neurobiological basis to beauty.

As opposed to...?


"As opposed to…?"

Blank Slate creationism holds that the mind is infinitely plastic with no built-in structures. Suggesting that beauty might have a neurobiological basis is apostasy, heresy, and a violation of PC.

Don't believe me?

Ask Larry Summers

Even within blank slate theory, isn't *everything* in the human condition resultant of interaction betwixt brain and environment?

Alexei Sadeski,

"Even within blank slate theory, isn’t *everything* in the human condition resultant of interaction betwixt brain and environment?"

Two answers. First, in the classic "Blank Slate" formulation, the mind is simply an empty vessel that can be filled with anything. In other words, humans can (according to the "Blank Slate") be taught any standard of beauty or none at all. That's quite different from suggesting that humans are neurobiologically preprogrammed to look for, and appreciate beauty. The former is strictly a social construction imposed on individuals by society and culture. The latter is the biology of the mind.

The second answer is more complex. Apparently, quite a few people believe in the "Ghost in the Machine" (from S. Pinker). In this formulation of human nature, it is not the interaction of the physical brain and the environment that counts, but the interaction of the Ghost and the environment.

You're just bitching about Steven Pinker's strawmen. Anyway, the fact that a brain scanner lights up under certain conditions is not actually evidence against the Crazy Infinitely Plastic Blank Slate Hypothesis. Alexei is right, we now have some more proof that thinking happens at least partly in the brain. Pop some champagne.

Studies of sex selection, particularly those that show an affinity for symmetry seem to demonstrate a common basis for beauty.

The mind is an artifact of an ordered biological structure. Autonomic responses and basic instinct demonstrates programmed behavior and responses to stimulus. We are most definitely not blank slates although I agree we are quite maleable. Nurture vs Nature is a false dichotomy.


I agree that beauty reflects both nature and nurture. A good example, is the cultural interpretation of mountains. Historically, they were viewed as dark, dangerous, and evil places that should be avoided. Starting in the late 18th century, this worldview was inverted and mountains came to be seen as thrilling, beautiful, and fun.

Of course, there were real social, economic, and medical reasons for the shift. However, if our terrain preferences were strictly biological, we would (presumably) still see mountains the same way we did in 1500.

A minor counter corollary is the suggestion that the preference of open landscapes (and efforts to create them) reflects a neurobiological bias in favor of open land (presumably to avoid predators).


“As opposed to…?”

A social and cultural construct created by the oligarchy to divide, enslave, oppress, imprison, etc.

To be fair, standards of beauty are not fixed over time, place, culture, etc. The notion that all such standards are implicitly oppressive is absurd. However, a cultural component clearly exists and is not fixed. Conversely, there is apparently enough commonality in which constitutes beauty to suggest that a neurobiological basis exists as well.

Language provides a partial analogy. Clearly, human language are highly variable. Conversely, they appear to share certain basic structures suggesting a neurobiological basis. For example, 75% of human languages are SVO or SOV. Compsky is a notable advocate for the "innate" (neurobiological) view of language. Predictably, even though Compsky is a figure of the left (far left) he has been denounced for Anglocentrism and Eurocentrism.

Is Euler's formula a definition, or a theorem? That's what I always have a hard time getting my mind around.

Put another way, is there some standard most-simple-and-elegant way of defining/deriving complex exponentiation, so that the beauty becomes most apparent?

Eluer's formula is definitely a theorem. Of course, you can change the definitions so that it becomes a triviality (that's what my text book of Terminale --senior class in high school-- in France, and it was ugly), but with the real, historical, definitions of the numbers involved, it is not. Its proof is very simple by modern standards, yet a remarkable achievement of differential calculus.

It goes roughly like this. The number pi is defined as the Greeks did, as the ratio of the circumference of a circle by its diameter. I assume there is no problems with the definitions of 0 and 1, and even of i and the complex numbers. As for e, and more generally e^z (for z a real or a complex number), there are several
equivalent ways to define it that were known to Euler (and their equivalence as well was well known to it), and one of them is to define e^z as the limit of the series 1+z+z^2/2+z^3/6+z^4/24+... (the general term is z^n/n!). Mathematics of Euler's time (and even decades before him) was able to prove without difficulty that this series actually had a limit.
In particular, e = e^1 is defined as 1+1+1/2+1/6+1/24+..., and can be computed with arbitrary precision using this series.
The definition settled, it should become clear that Euler's formula is by no means a tautology. There seems to be a priori no relation between this number e and the number pi of geometry.

For the proof (at least, one proof), you need to consider the trigonometric functions sin and cos, defined using the geometry of the circle, which were known to the late greek and arab mathematicians. Using classical trigonometric formulas, proved long ago using geometric argument à la Euclide, it is possible to compute the derivative, in the sense of Newton and Leibniz, of sin and cos (which are cos and -sin respectively), and from this to deduce that both cos and sin are equal to their second derivative, and then with a little more work to deduce a power series expansions from them: sin z = z - z^3/6 + ... , cos z=1-z^2/2+.... From this and the definition of e^z it follows that e^(i x) = cos x + i sin x for x real, and for x=pi, using that cos pi = -1, sin pi =0 by the very geometrid definition of sin and cos, we get Euler's formula. That's not a hard proof (as compared, say, as many other incredible proofs due the same Euler, certainly the most prolific and arguably the greatest mathematician of all times), but still a beautiful marriage between ancient geometry and modern differential calculus.

Wow, that helps, thanks for taking the time to type this up!

In my not-so-great math education (American high school) I think we learned to define the exponential as the inverse of the natural logarithm (and we never learned anything about complex logarithms). Later I learned about the Taylor series expansion, but I never thought of this as a definition. So there was this hole in my education, and when I learned Euler's formula I was never quite sure where it came from.

I also found this Wikiepdia article, which I found pretty helpful for understanding the exponential function:

I think of Euler's identity as little more than a cheap trick.

On the other hand, the first time I saw a Newton iteration in action (calculating a square root) I thought it was magical. That was forty years ago. I still find quadratic convergence magical. I have the same reaction to a merge sort. The SVD is also pretty awesome.

It is not efficient but one can calculate the inverse of matrix A by the following matrix iteration (if the first approximation X[0] is good enough):

X[i+1] = 2*X[i] - X[i]*A*X[i]

Euclid's proof that there infinitely many primes is beautiful.

undergrad analysis and abstract algebra was as far as I went, but I was awestruck over unexpectedly simple proofs and counter-intuitive results.

instantly relating to the title of this post is a simple reminder of just how geeky we are.

"Newton’s Binomial is just as beautiful as Milo’s Venus.
The issue is that few people realise this.

Whooosh – whoooooosh – whooooooooosh
(The wind outside)."

Alvaro de Campos
(Pen name of Fernando Pessoa [])

I agree with everyone and note that both mathematicians and physicists have repeatedly agreed on all polls taken: this is the most beautiful and impressive of all equations/identities of them all. Euler himself considered it to prove the existence of God. I have no comment on that, but I note that if one assumes this equation with all its proper definitions, one can basically derive most of the rest of standard mathematics to a pretty high level.

I don't know. Perhaps Eulers e^{i x} = cos x + i sin x has by now become as well known enough to NYT as e = m c ^ 2; but is that really as beautiful as, say, Galois theory, theta functions/ modular forms, 20th century unification of arithmetic and geometry (and analysis) ?

On the other hand, Atiyah is one of the co-authors, so what do I know?

I love S = k log W.

But I have a personal attachment to one I derived for supply and demand:

dD/dS = (1/k) (D/S)

[The equation gives you supply and demand diagrams if you solve it holding either D or S constant i.e. exogenous.]

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